Abstract:
We present an algorithm for reducing a given deterministic finite
state automaton (DFA) into a hyper-minimized DFA, which may have
fewer states than the classically minimized DFA. The price we pay
is that the language recognized by the new machine can differ from
the original on a finite number of inputs. These hyper-minimized
automata are optimal, in the sense that every DFA with fewer
states must disagree on infinitely many inputs.

Within a class of finitely differing languages, the
hyper-minimized automaton is not necessarily unique. There may
exist several non-isomorphic machines using the minimum number of
states, each accepting a separate language finitely-different from
the original one. We will show that there are large
structural similarities among all these smallest automata.

Abstract:
In recent years, straight-line programs (SLPs) turned out to be
a convenient formalism for investigating algorithms on
compressed string data. An SLP is a context-free
grammar that generates exactly one string w. Since the
length of the generated string w may grow exponentially with
the size of the SLP, the latter may be seen as
a compressed representation of w. The output of many
practical compression algorithms, like for instance those of the
Lempel-Ziv
family, can be considered as SLPs. The talk will
give an overview on recent complexity results for
algorithmic string problems, where the input string is given
compressed as an SLP. In particular, upper and lower
complexity bounds for word problems for
various language classes will be considered.
Also applications in combinatorial group theory will be presented.

Abstract:
Ciliates are an old and diverse group of unicellular eukaryotes. One
of their unique features is that they have two types of functionally
different nuclei, each present in multiple copies in each cell:
micronuclei and macronuclei. The difference between the micronuclear
and the macronuclear genome is striking, especially in Stichotrichs,
on which we concentrate in this talk. While macronuclear genes are
contiguous sequences placed in general on their own molecules,
micronuclear genes are placed on long chromosomes, interrupted by
stretches of non-coding material. Even more striking is that the
micronuclear genes are split into several blocks (up to 44 of them
in certain species), with the blocks arranged in a shuffled order,
separated by non-coding material. Some blocks may even be inverted!
At some stage during sexual reproduction, ciliates assemble the
blocks in the orthodox order to yield the transcription able
macronuclear gene. In this process, ciliates make use of some short
specific sequences at the extremities of each MDS in the same way as
pointers are used in computer science. Indeed, each coding block
ends with a short sequence of nucleotides that is repeated in the
beginning of the coding block that should follow it in the orthodox
order.

We will give in the talk an overview of some of the research done on
the mathematics of gene assembly in ciliates. Some of the techniques
are from formal language theory, others are from combinatorics and
graph theory, while others are from theory of computability. We
discuss several models stemming from the process of gene assembly,
based on permutations, strings, graphs, or formal languages. We
discuss results on closure properties of various language families,
invariants, completeness, parallelism, and even computing paradigms
based on gene assembly. We will also discuss a number of open
problems in the area.

Abstract:
Being sequential --- or input deterministic --- is a fundamental
property of machines, understood here as automata with output, the
output being taken in an arbitrary semiring of multiplicities. To some
extent, it amounts to say that the model can readily translate into a
physical device. Whether a function realised by a given finite
automaton can be realised by an equivalent sequential one is thus a
natural as well as an essential question: a negative answer somehow
implies that there is no finite realisation of the machine or that
performing the computations described by the model may be very costly.

Every finite automaton may be seen as an automaton with output, as it
was the case at the dawn of the theory with the Moore or Mealy
machines and the first characterisation of sequentiality, due to
Raney, goes back to the late fifties.

In this talk, I will try to survey and organise the known answers to
the question of sequentiality. We shall see that depending on the
semiring of multiplicities, the question goes from obvious to open,
that the answer goes from yes to undecidable. One may note that two
distinct complexity functions then arise, according to whether one
wants just to know the answer or to compute the equivalent sequential
automaton (in case it exists).